Arc-intersection queries amid triangles in three dimensions and related problems

Let T be a set of n triangles in 3-space, and let \Gamma be a family of algebraic arcs of constant complexity in 3-space. We show how to preprocess T into a data structure that supports various \emph{intersection queries} for query arcs \gamma \in \Gamma, such as detecting whether \gamma intersects any triangle of T, reporting all such triangles, counting the number of intersection points between \gamma and the triangles of T, or returning the first triangle intersected by a directed arc \gamma, if any (i.e., answering arc-shooting queries). Our technique is based on polynomial partitioning and other tools from real algebraic geometry, among which is the cylindrical algebraic decomposition.

Our first result is an O^*(n^{4/3})-size data structure that answers a query in O^*(n^{2/3}) time. Next, we devise an O^*(n)-size data structure that answers a query in O^*(n^{4/5}) time. Incorporating this structure at the leaf nodes of the main structure reduces its overall size to O^*(n^{11/9}), without affecting its query time which remains O^*(n^{2/3}). We also present a data structure that provides a trade-off between the query time and the size of the data structure. For example, if \Gamma is a family of algebraic arcs defined by t > 1 real parameters, increasing the storage to O^*(n^{3/2}) decreases the query time to O^*(n^{\rho}), where \rho=\max {1/2, (2t-3)/3(t-1)} < 2/3. We also show that this query time can be further improved for circular query arcs.

Our approach can be extended to many other intersection-searching problems in three and higher dimensions. We  exemplify this versatility by giving an efficient data structure for answering segment-intersection queries amid a set of spherical caps in 3-space, and we lay a roadmap for extending our approach to other intersection-searching problems.

Joint work with Pankaj Agarwal, Boris Aronov, Matya Katz, and Micha Sharir.